{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2021:3VKQIOYWSGMGVYXPXEL3DBKT5B","short_pith_number":"pith:3VKQIOYW","schema_version":"1.0","canonical_sha256":"dd55043b1691986ae2efb917b18553e85c20d9c2e8dcabec5e08567ac491d4af","source":{"kind":"arxiv","id":"2111.08410","version":3},"attestation_state":"computed","paper":{"title":"Thoughts on the Consistency between Ricci Flow and Neural Network Behavior","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.IT","math.IT"],"primary_cat":"cs.LG","authors_text":"Jun Chen, Tianxin Huang, Wenzhou Chen, Yong Liu","submitted_at":"2021-11-16T12:23:09Z","abstract_excerpt":"The Ricci flow is a partial differential equation for evolving the metric in a Riemannian manifold to make it more regular. On the other hand, neural networks seem to have similar geometric behavior for specific tasks. In this paper, we construct the linearly nearly Euclidean manifold as a background to observe the evolution of Ricci flow and the training of neural networks. Under the Ricci-DeTurck flow, we prove the dynamical stability and convergence of the linearly nearly Euclidean metric for an $L^2$-Norm perturbation. In practice, from the information geometry and mirror descent points of"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2111.08410","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2021-11-16T12:23:09Z","cross_cats_sorted":["cs.IT","math.IT"],"title_canon_sha256":"61afaf1756dd763dac047d54664b166018ede598360acd5af212f9ccf475a316","abstract_canon_sha256":"d9aebf4bc76c196e0be5d3c18c8e1ba3d1be8f34dec33f3dd4fc05c1199c82be"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T03:57:29.924126Z","signature_b64":"HW0IqNvc0TXBZsEubaXo5kHflhKZyf3dUw8lTGk1zWlQT9dH/hZnQOeZ/RkTEVrjZO/BMIHcXbmaFAE+S4GEBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"dd55043b1691986ae2efb917b18553e85c20d9c2e8dcabec5e08567ac491d4af","last_reissued_at":"2026-07-05T03:57:29.923752Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T03:57:29.923752Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Thoughts on the Consistency between Ricci Flow and Neural Network Behavior","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.IT","math.IT"],"primary_cat":"cs.LG","authors_text":"Jun Chen, Tianxin Huang, Wenzhou Chen, Yong Liu","submitted_at":"2021-11-16T12:23:09Z","abstract_excerpt":"The Ricci flow is a partial differential equation for evolving the metric in a Riemannian manifold to make it more regular. On the other hand, neural networks seem to have similar geometric behavior for specific tasks. In this paper, we construct the linearly nearly Euclidean manifold as a background to observe the evolution of Ricci flow and the training of neural networks. Under the Ricci-DeTurck flow, we prove the dynamical stability and convergence of the linearly nearly Euclidean metric for an $L^2$-Norm perturbation. In practice, from the information geometry and mirror descent points of"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2111.08410","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2111.08410/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"2111.08410","created_at":"2026-07-05T03:57:29.923810+00:00"},{"alias_kind":"arxiv_version","alias_value":"2111.08410v3","created_at":"2026-07-05T03:57:29.923810+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2111.08410","created_at":"2026-07-05T03:57:29.923810+00:00"},{"alias_kind":"pith_short_12","alias_value":"3VKQIOYWSGMG","created_at":"2026-07-05T03:57:29.923810+00:00"},{"alias_kind":"pith_short_16","alias_value":"3VKQIOYWSGMGVYXP","created_at":"2026-07-05T03:57:29.923810+00:00"},{"alias_kind":"pith_short_8","alias_value":"3VKQIOYW","created_at":"2026-07-05T03:57:29.923810+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2501.12116","citing_title":"Efficient PINNs via Multi-Head Unimodular Regularization of the Solutions Space","ref_index":15,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/3VKQIOYWSGMGVYXPXEL3DBKT5B","json":"https://pith.science/pith/3VKQIOYWSGMGVYXPXEL3DBKT5B.json","graph_json":"https://pith.science/api/pith-number/3VKQIOYWSGMGVYXPXEL3DBKT5B/graph.json","events_json":"https://pith.science/api/pith-number/3VKQIOYWSGMGVYXPXEL3DBKT5B/events.json","paper":"https://pith.science/paper/3VKQIOYW"},"agent_actions":{"view_html":"https://pith.science/pith/3VKQIOYWSGMGVYXPXEL3DBKT5B","download_json":"https://pith.science/pith/3VKQIOYWSGMGVYXPXEL3DBKT5B.json","view_paper":"https://pith.science/paper/3VKQIOYW","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2111.08410&json=true","fetch_graph":"https://pith.science/api/pith-number/3VKQIOYWSGMGVYXPXEL3DBKT5B/graph.json","fetch_events":"https://pith.science/api/pith-number/3VKQIOYWSGMGVYXPXEL3DBKT5B/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/3VKQIOYWSGMGVYXPXEL3DBKT5B/action/timestamp_anchor","attest_storage":"https://pith.science/pith/3VKQIOYWSGMGVYXPXEL3DBKT5B/action/storage_attestation","attest_author":"https://pith.science/pith/3VKQIOYWSGMGVYXPXEL3DBKT5B/action/author_attestation","sign_citation":"https://pith.science/pith/3VKQIOYWSGMGVYXPXEL3DBKT5B/action/citation_signature","submit_replication":"https://pith.science/pith/3VKQIOYWSGMGVYXPXEL3DBKT5B/action/replication_record"}},"created_at":"2026-07-05T03:57:29.923810+00:00","updated_at":"2026-07-05T03:57:29.923810+00:00"}